Suppose $X_i \overset{iid}{\sim} N_m(0, \Sigma)$ for $i=1, \dots, n$ for the case that $$\Sigma = \sigma^2((1-\rho)I_m + \rho \mathcal{1}\mathcal{1}^\prime),$$ where $\mathcal{1} \in \mathbb{R}^m$ is a vector of ones. Letting $S$ be the standard sample covariance matrix, (that is, $S = \frac{1}{n}\sum_{i=1}^n (X_i - \bar{X})^T(X_i -\bar{X})$) Muirhead (1985, p114) shows that the MLE for $\rho$ is given by $$ \hat\rho = \frac{\mathcal{1}^\prime S \mathcal{1} - trace(S)}{(m-1)trace(S)}.$$
I am wondering if anyone knows of a reference where the distribution, or asymptotic distribution, of this MLE is established?
